Algebra 2 / Unit 6: Rational Functions
6.5 Solving Rational Equations
Notes
- Multiply every term by the LCD to clear the fractions.
- Solve the resulting equation.
- Check: any answer that makes an original denominator 0 is extraneous.
- Example: $\displaystyle \frac{4}{x} = \frac{2}{x - 1} \Rightarrow 4(x - 1) = 2x \Rightarrow x = 2$.
- Work problems: rates add. If one worker takes $a$ hours and another $b$ hours, together $\displaystyle \frac{1}{a} + \frac{1}{b} = \frac{1}{t}$.
Practice
Solve $\displaystyle \frac{3}{x} + \frac{1}{2} = 2$.
Multiply by $2x$: $6 + x = 4x$, so $x = 2$.
Solve $\displaystyle \frac{x}{x - 3} = \frac{3}{x - 3} + 2$.
Multiplying by $x - 3$ gives $x = 3 + 2x - 6$, so $x = 3$. That makes the denominator 0, so it is extraneous.
One hose fills a pool in 6 hours. A second hose fills it in 3 hours. How long do they take together?
$\displaystyle \frac{1}{6} + \frac{1}{3} = \frac{1}{2}$ pool per hour, so 2 hours.
Ready for more? Take the Unit 6 test.